Construction of distance matrices from single-cell chromosome imaging data

For this work, we have examined experimental single-cell chromosome imaging data at three different resolutions, each of which reflects different levels of structural organization of the 3D genome at different genomic length scales. The single-cell chromosome imaging data obtained from Bintu et al. [37]. has the highest genomic resolution of 30 kb. Here, the authors traced the conformations of multiple ~ 2–2.5 Mb spanning chromatin regions of chr21, partitioned into consecutive 30 kb segments in single-cells across different cell-types (e.g., IMR90 –human fetal lung fibroblast cells, K562 –human erythroleukemic cells, A549 –human lung adenocarcinoma cells, and HCT116 –human colon cancer cells) during different physiological and perturbed conditions. The second set of IMR90 chromatin tracing data analyzed here is obtained from Su et al. [38], where the single-cell conformations of chr21 non-repetitive region were traced using consecutive 50 kb bins. In addition, that dataset also provides chr2 conformations captured at the same 50 kb resolution, however with an interval of 250 kb between two genomic regions, due to very large size of human chr2. Both the Bintu et al. and Su et al. data capture chromatin conformations at a comparable resolution (30 and 50 kb respectively), differing only in the size of the genomic region analyzed. Whereas the Bintu et al. data primarily represents the structural heterogeneity at the level of specific TADs and sub-TADs, the Su et al. data captures the heterogeneity of almost the whole chromosome. Finally, we also analyzed chromosome structural heterogeneity of the whole active and inactive chrX of IMR90, obtained from Wang et al. [36]. In this dataset, TADs are imaged as the basic subunits along the chromosome, so the resolution of this data is lower (ranging from 440 kb to 7.88 Mb) and reports on the conformations of compartments and the whole chromosome rather than the within-TAD level.

For all these different datasets, the authors reported the single-cell conformations of the chromosomes in a 3D Cartesian coordinate system where each basic genomic region (e.g., 30 or 50 kb genomic region or TAD; from here termed as bin) has X-, Y-, Z- coordinate values. However, in some single cell structures the 3D positions of some of the bins are missing due to technical difficulties. Therefore, to decrease the effect of missing data, we consider only structures with more than ~ 90% measured bins, and the values for missing bins in those structures are assigned by nearest-neighbor interpolation. The 3D positions of chromosomes are also associated with the global rotational and translational variations which might affect underlying statistical analysis. To resolve this issue, we convert each 3D structure into a 2D symmetric distance matrix by calculating the pairwise Euclidean distance between all the genomic bins of that chromosome structure. For example, if a particular chromosome has m genomic bins and a total of N copies originating from N single cells, then we will have N symmetric pairwise distance matrices each of size m×m, where d_ij represents the Euclidean distance between i^th and j^th genomic bins. Further, the spatial distance can be converted into contacts by applying a genomic resolution specific thresholding cutoff. However, for our analyses, we use distance matrices instead of contact matrices, unless otherwise explicitly mentioned.

Generating chromosome structural landscapes using a statistical analysis technique

We next center all the single-cell matrices around zero by subtracting the median distance matrix from each of them, h_ij = d_ij−median(d_ij). Once we obtain the median centered single-cell distance matrices, we calculate the pairwise dot product between each of those single-cell matrices, such as, , where h^α and h^β are median centered distance matrices obtained from single cells (α, β∈{1,…,N}), respectively. For each pair of single-cells, this step compares the deviation from the median between the two structures and yields a single number. In the case of two structures having similar deviations, this dot product would yield a strong positive value and vice versa. The pairwise dot product comparison information between all possible pairs of the single cells is then represented in a symmetric matrix format, termed as pairwise structure comparison matrix, where each element of the matrix represents the pairwise comparison information between a specific pair of chromosome structures. Next, we perform PCA on the pairwise structure comparison matrix and project the single-cell conformations in 2D space using the first two principal components (PCs). This PC projection is termed the chromosome structural landscape.

Quantification of chromosome structural ensemble variation using a network-based similarity index

To quantify the variation associated with the chromosome structural ensemble, we perform a network-based similarity calculation on the pairwise comparison matrix (Fig 1C). Similar approaches have been used to study cellular network heterogeneity associated with cancer and cellular differentiation [46–48]. In this approach, we first calculate Pearson’s correlation coefficient (PCC) between each pair of rows of the comparison matrix, where c_ij denotes PCC between i^th and j^th rows of the pairwise similarity matrix. As PCC values range between -1 and 1, next we scale the PCC values in the range of 0 and 1, w_ij = 0.5*(1+c_ij). Once we have the rescaled pairwise PCC value matrix, we construct a weighted network from that where each node represents a single chromosome structure or conformation and the edges between them denotes rescaled PCC values. However, we only consider the edges having weights above a certain threshold, representing only connections between highly similar structures. This threshold is calculated by taking 50^th percentile of all the edge weights of the network (see Results for an analysis of the impact of this threshold choice). With the constructed network, we calculate the structure similarity index (SSI) of each node S_i, using a formula based on the Shannon-Jayne network entropy [46]. where N(i) represents the neighborhood set of the node i and k_i denotes the cardinality of that set. p_ij represents the proportionality of the edge weight between nodes i and j compared to all the edges associated with node i.

Network entropy is a commonly applied concept in the analysis of heterogeneity vs. continuity in biological networks [49]. Conceptually, entropy measures the degree of randomness in a system, or the extent to which all possible outcomes are sampled. Entropy in DNA sequence analysis measures whether all 4 DNA bases are represented uniformly in a motif position (high entropy) or some bases are strongly preferred in certain positions (low entropy) [50,51]. Network entropy in signaling or gene expression networks captures whether the network is more continuous or modular [46,52,53]. Similarly, here this metric will reveal whether the landscape of structures is highly connected with all intermediates sampled vs. more heterogeneous with disconnected structures. This index S_i is bound between 0 and 1. In our system, the resulting values are compressed near a value of 1, so for easier visual presentation of values, we transform them by subtracting 0.99 from each SSI and then multiplying by 1000. For our presented analyses, we calculate the median SSI across all nodes in the landscape and call this the ensemble SSI. Specifically, to normalize sample size between conditions, we iteratively resample a subset of nodes from each condition, then plot the distribution of median SSI values from the resampled node sets. Based on the distribution of the median SSI values, we characterize the heterogeneity of the chromosome structural landscape. An SSI distribution with higher mean is considered as having a smooth continuous structural landscape while a distribution with relatively lower mean would have more discrete, disconnected structures. Interpretation of the SSI distribution trends have been discussed in detail in the Results section.

Calculating chromosome structure variability for each genomic bin

To characterize the variability of structure at each genomic bin along the chromosome, we use a modified version of the network-based structural similarity calculation approach, discussed in the previous subsection (Fig 1D). In this step, for each single-cell structure, we first construct a network based on its distance matrix, where each node is a single genomic bin and the edges between them represent the spatial distances. We then threshold the network using an experimentally-derived distance cutoff (330 nm for 30 kb resolution [41] and 500 nm for 50 kb resolution [38]), where edges with spatial distance higher than the cutoff are discarded from the analysis. Next, we calculate the bin similarity index of each of the nodes (bins) in the network using the same formula discussed in the previous section, except here using genomic bins as nodes rather than whole single-cell structures and spatial distances as edge weights rather than structure correlation coefficients. We now have a bin similarity index value for each genomic bin in each single cell structure. To quantify the variability of structural positions occupied by a given bin across all structures, we then calculate the standard deviation of bin similarity index values across all single cell structures for a given bin. We call this measure the bin variability. A higher standard deviation corresponds to a more dynamic or variable genomic bin.

Characterization of chromatin states using epigenetic profiles

In order to characterize chromatin states of the IMR90, K562 and A549 cells, we downloaded a total of eleven histone marks–H3K4Me1, H3K4Me2, H3K4Me3, H3K9Ac, H3K27Ac, H4K20Me1, H3K36Me3, H3K27Me3, H3K9Me3, H2A.Z and DNase for all the three cell types from ENCODE repository (S1 Table) [54]. Next, we used chromHMM software [55] to annotate genomic regions of desired resolutions into five chromatin states using a combinatorial pattern of the eleven histone marks. Based on the combinatorial histone marks, we defined chromatin states as–Active state 1, Active state 2, Inactive Poised, Repressed Polycomb and Heterochromatin (S1 Fig). Briefly, if an inferred state shows strong enrichments for active marks such as H3K4Me1/2/3, H3K27Ac, H3K9Ac and DNase, we term that state as an Active state. Further, depending on the level of enrichment of those different active marks, we characterize them into Active state 1 and Active state 2. We assign the Heterochromatin label to the state for which we observe strong enrichment of H3K9Me3 and for the Repressive Polycomb assignment, we look for strong H3K27Me3 and a mix of H3K9Me3 and H4K20Me1 marks. And the rest of the states are assigned the Inactive Poised level.

Polymer simulation of chromatin segments based on underlying chromatin states

To test the contribution of epigenetic states to chromosome structure heterogeneity, we perform polymer simulations of a specific chromatin segment with the interactions defined based on their epigenetic states. In recent years, polymer simulation-based studies have shown how the compartmental profile or the underlying epigenetic state of the genomic regions can give rise to the population-averaged structural organization of chromatin at different resolutions [9,56,57]. Here, we represent a ~ 1.95 Mb chromatin region as a chain of beads using a bead-spring polymer model. The model system consists of 1950 beads where beads represent genomic regions of 1 kb. Each bead is assigned a specific chromatin state as described above. In addition to that, we also model loops within the specific chromatin regions based on publicly available high-resolution Hi-C derived chromatin loop sets from Rao et al. [58]. For each loop, we connect the loop anchors using a harmonic constraint. The potential energy function of the polymer model is described below, where U_B(r) corresponds to bond stretching potential, modeled by finite extensible nonlinear elastic potential given by, where is spring constant and R₀ = 1.6σ is the equilibrium bond length.

We use Lennard-Jones potential to model steric repulsion and attraction between the beads of different chromatin state with different parameters. For example, the potential between a bead of chromatin state A and that of B is represented below, where ϵ_AB denotes interaction strength between bead types A and B (S2 Table) and σ refers to the size of each bead.

In case of loops, we use harmonic potential between each of the loop anchors U_L(r), represented by, where is spring constant and is the equilibrium loop bond length.

Finally, in order to model the repulsive interaction between the chromatin beads and the simulation box boundaries, we use Lennard-Jones potential where ϵ_W = 1.0 denotes the interaction strength between boundary walls and chromatin beads irrespective of their types.

The LAMMPS molecular dynamics package is used to perform polymer dynamics simulations [59]. All the simulations are performed in terms of Lennard-Jones reduced units with τ for time, σ for length and ϵ for energy. To mimic the viscoelastic nature of the nucleoplasm, Langevin dynamics with a damping coefficient of γ = 1.0τ is used to maintain the temperature at T = 1.0. We then run the simulation for a total of 1,000,000 steps with a timestep of dt = 0.000001τ. Followed by that, the timestep dt is increased to 0.01τ and a production run of a total 2,000,000 steps is performed and the conformations are saved at intervals of 1,000 steps. We repeated this process for 20 times, each time with different initial conformation and random seeds.

Initial conditions and Equilibration

The chromatin fiber is initialized as a self-avoiding worm like chain using the python package PolymerCpp [60] and is placed in a box of size L = 35σ with fixed repulsive boundaries. We then allow the polymer to equilibrate inside the box using four consecutive phases. In the first phase, a soft potential is used to remove overlap in the polymer for 400,000 steps with a dt = 0.01τ. where A is pre-factor, varied gradually from 0 to 100ϵ and .

For the rest three subsequent phases, we switch to potential to Lennard-Jones potential and run the simulation for 400,000, 500,000 and 500,000 steps with timesteps 0.000001τ, 0.0001τ, and 0.01τ respectively. where ϵ_G = 1.0 denotes the interaction strength between chromatin beads irrespective of their types.

Loop extrusion simulation of chromatin segments based on underlying RAD21 binding sites

To test the contribution of loop extrusion dynamics to the variability present in chromosome structural ensemble, here we perform loop extrusion polymer simulations of a specific chromatin segment according to the method described in Fudenberg et al. [2]. For these simulations, we define the domain boundaries based on RAD21 peaks present in the desired chromatin segment. Here, we model a 2 Mb spanning chromatin region as a 2000 beads monomer chain where beads represent genomic regions of 1 kb. All the simulation parameters are kept same between different cell types except the position of the boundary elements (here RAD21 peaks) and their strengths. The strength of boundary elements is defined in terms of boundary permeability parameter in the models which represents the boundary insulation. For each cell-type, these parameter values are obtained by calculating the relative strength of the RAD21 peaks. The RAD21 peaks for the IMR90 and K562 were downloaded from the ENCODE repository [54].

Analysis of nucleolus-associated domain (NAD) mapping data

In order to characterize the nucleolar association of the IMR90 chromosomal regions, we downloaded the NAD mapping data of the proliferating IMR90 cells from Dillinger et al. [61]. This microarray data contains log2 normalized ratio of nucleolus-associated and genomic DNA at specific probe locations. For our analysis, we bin that log2 normalized ratio data at 10 kb resolution using multiBigwigSummary functionality of the deepTools software [62].

Chromosome structural landscape reveals major features of the structural ensemble

We first apply our statistical analysis technique to the single-cell chromosome structures measured for a 2 Mb region of IMR90 chr21 (28–30 Mb) (Fig 2A) and produce a chromosome structural landscape (Fig 2B). We observe that the first major source of variation corresponds well to the compactness of the chromosome structures, which can be quantified using the radius of gyration, R_g (S2A Fig). We calculate the R_g of the single-cell structures from their experimentally obtained 3D conformations. When we compare the R_g values of the structures with their PC1 values (from here on we refer to the PC1 projection as “PC1”), we observe that the structures with more negative PC1 projections are highly compact with smaller R_g values while structures with more positive PC1 values are less compact with higher R_g scores. With increasing R_g, the structural divergence also increases, with individual structure points becoming spread further along PC2 and also more spread out from each other.

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Fig 2. The chromosome structural landscape orders single-cell structures on a continuum based on their radius of gyration followed by domain reorganization.

a) Population-averaged distance map of IMR90 chr21:28–30 Mb at 30 kb resolution. b) Chromosome structural landscape of IMR90 chr21:28–30 Mb chromatin region, generated as described in the Methods section. Each point represents a single-cell structure. Points are colored according to the radius of gyration (R_g) of the corresponding structures (dark blue–smaller R_g and dark red–larger R_g). The percent of variation explained by each PC is shown in parentheses. c) Structures are first divided into five groups based on the PC1 ordering (which correlates highly with R_g). Structures belonging to groups 2, 3 and 5 are shown. Within each group structures are further divided into five subgroups based on their PC2 ordering. Average structures within each subgroup are shown.

https://doi.org/10.1371/journal.pcbi.1010392.g002

Along PC2, we observe that the structure ordering follows a progressive reorganization of the domains within the region and captures several subclusters of structures (Fig 2C). As we move from negative to positive PC2, we observe a shifting of a boundary (weakening of one boundary and gaining of another) that leads to the change in the structure of the domain (S2B Fig). This phenomenon can be seen in different regimes of R_g, showing the existence of domain reorganization within both compact and open structures (S2B Fig).

When the same approach is applied to different chromosomes and cell-types at different resolutions, we found a consistent ordering of the dynamic modes of the structural landscape (S2C, S2D, S3 and S4 Figs). Although the major dynamic variation remains the R_g ordering for all the different situations, the reorganization of domains along PC2 is dependent on the length of the genomic region as well as the local structural properties.

Ensemble structure similarity index captures cell-type- and chromosome-specific features of the chromosome structural ensemble

To compare the single cell structural heterogeneity between different cell types, we constructed a joint structural landscape using the 28–30 Mb chr21 region data from three cell types–IMR90 (lung fibroblast), K562 (leukemia) and A549 (lung adenocarcinoma) collectively. In this joint structural landscape we can visualize the cell-type-specific variation present in structural ensemble (Fig 3A). To quantify the differences between cell types, we divide the landscape in 10 equally sized regimes along PC1 (since PC1 correlates with R_g, different PC1 regimes correspond to different R_g values) and calculated the proportion of different cell types within each regime. We observe that K562 shows a higher proportion of chromosome structures in lower R_g regimes and has an overall more compact structural ensemble compared to A549 and IMR90 (Fig 3A and 3B).

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Fig 3. Network based similarity score quantifies cell-type- and chromosome-specific variability present in chromosome structural ensemble at different resolutions.

a) Jointly constructed chromosome structural landscape of genomic region chr21:28–30 Mb for cell types IMR90, K562 and A549. The only difference between individual and joint landscape is that all the different cell-type structures are considered together while calculating pairwise structure comparison matrix. Points are colored based on the cell types. Further, the landscape is divided into ten equal sized bins (or regions) based on PC1 values, where the bins on the left represent lower R_g regions and bins on the right are with higher R_g. b) Relative proportion of IMR90, K562 and A549 single-cell structures present in each bin represented in a). c) Chromosome structural ensemble SSI distributions for different 2 Mb chromosome regions from different cell-types and conditions, obtained from Bintu et al. [37]. The SSI value of the structural ensemble is calculated using a network entropy based approach. For each structural ensemble (size of ensemble indicated as n), 200 single-cell structures are selected randomly (except IMR90 S 28–30 Mb where we select 150 structures for resampling), and the median of the SSI values of the nodes are calculated. This step is repeated 1000 times and values are plotted as a distribution. d) Chromosome structural ensemble SSI distributions for active and inactive IMR90 chrX at TAD resolution, obtained from Wang et al. [36]. 50 single-cell structures are selected randomly each time for resampling. e) Chromosome structural ensemble SSI distributions for different IMR90 chromosomes and conditions at higher resolution, obtained from Su et al. [38]. Here, for resampling 300 single-cell structures are selected randomly each time.

https://doi.org/10.1371/journal.pcbi.1010392.g003

While the relative R_g shift between cell types is straightforward to observe and quantify, other features of the chromosome structural landscape are more difficult to compare visually. Therefore, we devised a network-based measurement to quantitatively compare the structural landscapes, as discussed in the Methods section. According to this approach, a network with similar edge weights will exhibit a higher ensemble SSI value and vice-versa. In terms of the chromosome structural ensemble, a set of structures that smoothly transition across the entire conformational space will have a highly connected structural landscape and thus high ensemble SSI (Fig 1A, blue line). On the other hand, if structures coalesce around discrete states (Fig 1A, red line), the discontinuity in the structural ensemble will be higher, and the resulting network will have a larger variation in of edge weights, and a low ensemble SSI score will be produced.

We first applied the network-based similarity quantification approach to structures of 2 Mb regions of chr21 (30 Kb resolution) in different cell-types and conditions obtained from Bintu et al. [37]. There is variation in the sample size (i.e., the number of conformations) obtained for different experiments, so we relied on a resampling approach to make the comparison consistent. Based on this approach, when we compared the 28–30 Mb genomic region from three different cell-types analyzed above, we find that IMR90 has the highest ensemble SSI, followed by K562 and then A549 with the lowest ensemble SSI (Fig 3C). Thus, this region in IMR90 cells has the highest continuity across different structures while the corresponding distribution in A549 cells exists as a relatively discrete set of structures.

To evaluate our interpretation of the ensemble SSI result in an orthogonal fashion, we performed hierarchical clustering of the 3D structures obtained from IMR90 and K562 cells based on a structure similarity metric, Q, as represented in Cheng et al. (S5 Fig). At the same threshold, both cell types showed the same number of structure clusters. However, in IMR90, the vast majority of structures fall into one of these clusters, corresponding to our high ensemble SSI value (S5A and S5B Fig). In K562, the structures are more evenly split into the different clusters, corresponding to our low ensemble SSI value (S5C and S5D Fig). Thus, our ensemble SSI metric indeed captures the degree of continuity within the structural ensemble. By using ensemble SSI rather than a hierarchical clustering approach, we avoid the need to arbitrarily select a cutoff or number of clusters. We further assessed the dependence of the ensemble SSI score on the number of cells measured (S6 Fig). We observed that with an increase in the number of cells, the difference in ensemble SSI between cell-types follows the same trend, but the absolute difference becomes smaller (S6B Fig). It is important to note that as we increase the number of cells, we are including more low-quality measurements (cells with more missing data) and this may also contribute to a broader ensemble SSI distribution. We then examined how robust the ensemble SSI distribution is to changes in the correlation matrix cutoff chosen (S6C Fig). Using a cutoff lower than our initially chosen 50^th percentile produces similar results, but increases variability as more non-significant connections between structures are included. A high cutoff, on the other hand, can obscure the results by breaking a heterogeneous structure landscape into too many disconnected networks. Within each sub-network, there is high continuity, so this results in an artificially high SSI being reported for landscapes that are in fact heterogeneous. We conclude that a 50^th percentile correlation matrix cutoff excludes noisy connections without creating disconnected subnetworks (see S6C Fig for extended discussion). As a final technical test of our metric, we repeated the ensemble SSI analysis after removing extremely compact or large structures from each cell type to see whether those extreme conformations are disproportionately affecting the results (S7 Fig). We found that those extreme structures actually have little influence on the ensemble SSI calculation. Overall, we conclude that our ensemble SSI calculation is robust.

We next compared two different genomic regions of IMR90 chr21 using the ensemble SSI distribution. We found that although both are from same cell-type and same chromosome, the 28–30 Mb region has lower ensemble SSI values compared to 18–20 Mb region (Fig 3C). This difference indicates that the latter region has a higher degree of continuity in its structural ensemble. This 18–20 Mb region lies entirely in the B compartment (putative inactive, gene poor regions) while the 28–30 Mb region contains a majority of A compartment regions mixed with B compartment regions. To further investigate the potential connection between compact, inactive states and the continuity of the structural landscape, we compared the ensemble structural similarity of the active vs. inactive chrX structural ensembles (440 Kb to 7.8 Mb) (Fig 3D). We found that chrXi has higher ensemble SSI values compared to its active counterpart. This suggests that the chromosome with a highly compact global structure has a more continuous structural landscape compared to the more specific and discrete states of the open and active chromosome. (S8A, S8B and S8C Fig) [63].

When we analyzed the structural variation of the entire chr2 and chr21 q-arm at high resolution [38], we observed that chr21 has a higher ensemble SSI compared to chr2 (Fig 3E). To test whether the difference in length between chr2 and chr21 contributed to differences in the ensemble SSI values, we calculated the ensemble SSI of regions of different lengths selected from chr2. We observed that as the length increases the ensemble similarity decreases (S8D Fig). This effect likely stems from the increased number of possible conformations that a longer polymer can adopt, which corresponds to a sparser coverage of conformational space. We also found that ensemble similarity is lower in S/G2 phase compared to G1 phase for chr21. This result is consistent with the ensemble SSI values of 28–30 Mb IMR90 chr21 region structural ensembles from different cell-cycle stages, where the G1 phase has highest ensemble SSI followed by S and G2 phases, as shown in Fig 3A. One major difference between the chromosomes in the phases of the cell cycle is the addition of tethers between sister chromatids in S and G2 compared to G1 [64]. Increased constraints on the chromosome structure could then lead to more distinct structures and lower structural ensemble similarity.

Contribution of epigenetic properties to the cell-type-specific variation of chromosome structural ensemble

Building on the observation that regions with different A/B spatial compartment identity had different structural landscape SSI, we further examined the influence of local epigenetic state on the resulting structural landscape. Recent studies have shown the association of epigenetic states with the ensemble genome organization [28, 29, 65]. Fig 4A shows chromatin states along the 28–30 Mb genomic region in the three different cell types–IMR90, K562 and A549 (detailed steps in the Methods section). We observe that this region in IMR90 mostly has active marks while in K562 the region is mixed between repressive polycomb and heterochromatin marks as well as active marks. The A549 region shows a mixture of active, repressive, and poised marks. By qualitatively comparing this chromatin state data with the ensemble SSI distributions for these three cell types, we observe that the cell type with the more uniform epigenetic state (IMR90) has the highest ensemble SSI while cell types with mixed epigenetic states exhibit lower SSI.

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Fig 4. Epigenetic properties (histone modifications) contribute to the cell-type-specific variation in the chromosome structural ensemble.

a) Chromatin states of chr21:28–30 Mb genomic region from three different cell-types–IMR90, K562 and A549 at 30 kb resolution. The chromatin states are inferred based on 11 different histone marks for each cell-type by ChromHMM tool. b) Bin variability of the chr21:28–30 Mb region from IMR90, K562 and A549, where a higher value represents higher degree of structural variation of that genomic position across the population and vice versa. c) Population-averaged distance maps of IMR90 and K562 chr21:28–29.5 Mb genomic region from polymer simulations (see Methods) at 1 kb resolution. d) Chromosome structural ensemble SSI distributions for chr21:28–29.5 Mb from simulated IMR90 and K562. For resampling, 500 single-cell structures are selected randomly each time.

https://doi.org/10.1371/journal.pcbi.1010392.g004

In addition to the overall dynamic properties of the region, we sought to quantitatively compare local genomic bin epigenetic state with the degree of structure fluctuation experienced by that bin. We first calculated the bin variability for each cell type (Methods) which represents the structure variability of a particular genomic region across the population of cells (Fig 4B). We do not see a direct bin-by-bin correlation between epigenetic state and bin variability. Instead, we observe that a higher proportion of active states within the region tends to correspond to higher bin variability across the region while a larger proportion of inactive states, even if focused in one part of the locus, corresponds to restricted bin variability across the whole region. In this comparison, it is important to remember that we are comparing a population averaged epigenetic state with single cell structural fluctuation. It is possible that additional relationships could become evident if single cell variation in epigenetic state were considered.

To test our hypothesis that epigenetic state differences contribute to cell-type-specific variation in the chromosome structural ensemble, we performed a set of polymer simulations of the desired genomic region of 28–29.5 Mb at a resolution of 1 kb, as described in the Methods section. We generated the models for IMR90 and K562 separately by assigning interactions between the polymer beads based on their specific chromatin states (S9A Fig). Once we obtained the 3D conformations from the simulations, we constructed pairwise Euclidean distance matrices for each conformation and further quantified the structural ensemble using our structural similarity approach. For the resampling purpose, we combined all the snapshots from 20 independent simulations and then selected 500 conformations randomly from this set 1000 times. When we compared the simulated IMR90 ensemble SSI distribution with simulated K562 data, we found that the K562 structural ensemble shows a lower similarity, which is the same trend observed in our analysis of the Bintu et al. imaging data (Figs 3C and 4D). Since the only difference between cell types in these simulations was the encoded epigenetic state of the polymer beads, this result corroborates the idea that the epigenetic state is one factor that can contribute to the variation in chromosome structural ensemble.

Structure heterogeneity properties are associated with epigenetic state and boundary positions

We further evaluated the contribution of epigenetic factors to structural heterogeneity by analyzing single-cell structures of six genomic regions, each 2 Mb in length, obtained from IMR90 chr21 whole q-arm imaging data (Fig 5A) [38]. For simplicity, the regions will be denoted as S1 to S6 based on their starting genomic coordinates. The regions S1 (chr21:18–20 Mb) and S2 (chr21:28–30 Mb) were selected in order to match the two IMR90 genomic regions data analyzed in Bintu et al. The three regions S3, S5 and S6 were selected based on their epigenetic state similarity with epigenetic state of the chr21:28–30 Mb region in K562 cells (S9B Fig). To evaluate epigenetic state similarity, we designed a sliding window-based approach where we move a window of size 40 (2 Mb genomic region with 50 kb resolution) over the whole q-arm of chr21 and for each window compare the IMR90 epigenetic state with the K562 28–30 Mb region epigenetic state using binary comparison (S9C Fig). However, as the resolution of the Bintu et al. data (65 bins per 2 Mb region) is different from the Su et al. data (40 bins per 2 Mb region), we take the least common multiple of the two bin numbers and expanded each data to 250 bins by repeating each bin element the required number of times. Here, a higher comparison value shows higher similarity with the K562 epigenetic state. Among the three regions, S3 has the highest score, S6 has the second highest and S5 has the third highest one (S9C Fig). The region S4 was selected randomly.

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Fig 5. Evaluation of structural variation of regions with different epigenetic, genomic, and structural properties.

a) Population-averaged distance maps of 6 different genomic regions of IMR90 chr21 at 50 kb resolution obtained from Su et al. [38]. Locations of these 6 regions are indicated on the top right chromosome ideogram and labeled with colors used throughout the figure. b) Chromosome structural ensemble SSI distributions for the same 6 regions of IMR90 chr21. For resampling, 300 single-cell structures are selected randomly each time. c) HiCRatio score of each genomic bin of chr21 q-arm at 50 kb resolution (calculated with a 300 kb window size). The red points represent the boundary positions inferred by HiCRatio. d) Nucleolus association level of the same 6 regions of IMR90 chr21 [61]. For each genomic region, the 1 Mb upstream and downstream regions are also shown. A value greater than 0 represents higher levels of nucleolus association and vice versa. Colors in each panel correspond to chromosome locations indicated above panel b.

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When we quantified the variation in the structural ensemble of these six regions, we first observed that the S1 and S2 regions have ensemble SSI distributions that are consistent with the results from the same regions in the Bintu et al. data (Fig 5B). Next, we found that the S3 region has the lowest ensemble SSI distribution among all the six regions which is also consistent with the low ensemble SSI of the Bintu et al. K562 region. In addition, the ordering of S6 and S5 are also consistent with their similarity with K562 epigenetic state. Finally, the S4 region, which has a very low similarity score with K562 state, exhibits a very high ensemble SSI distribution. Again, this analysis supports the idea that epigenetic patterns contribute to the degree of variation present in the chromosome strucural ensemble.

In addition to the epigenetic states, we also identified domain boundaries along the whole q-arm of chr21 using the HiCRatio approach (with a 300 kb window size) [66]. Since this boundary calling software is designed for contact data, we first converted the single-cell chromosome structural distance data into contact data using a cutoff of 500 nm, as previously reported [38]. We found that regions with a higher number of strong domain boundaries exhibit lower ensemble similarity and vice versa (Fig 5C). For example, the S3 region, which has the lowest ensemble SSI, has three strong domain boundaries and one weak. On the other hand, regions like S1 and S4 which exhibit higher ensemble SSIs have fewer or weaker domain boundaries. This effect is visible across the entire chromosome, in that the average boundary strength in a given region is negatively correlated with the ensemble SSI of that region (S9 and S9E Fig).

Finally, we asked whether chromosome association with the nucleolus is associated with changes in the chromosome structural ensemble SSI (Fig 5D). The NAD data source and processing steps are described in the Methods section. We observe that S1 and S4, the regions with the highest structural ensemble similarity, are the most uniform in their NAD state: S1 is completely associated with the nucleolus, while S4 has almost no association with the nucleolus. As seen with epigenetic state, here we find that regions with uniform nucleolus association exhibit a higher continuity level in the structural ensemble. In contrast, S3 has the lowest ensemble SSI and also alternating levels of nucleolus association in the first half of the genomic region, followed by an unconstrained region. Thus, variation in constraints across the region is again associated with specific distinct structures and thus a lower ensemble similarity.

Loop extrusion alone does not reproduce experimentally detected chromosome structure variation patterns

Above, we observed that chromosome regions with a higher number or strength of domain boundaries showed less continuous structural landscapes and lower ensemble SSI values. Boundaries detected by approaches like HiCRatio can result from cohesin-mediated loop extrusion that is periodically blocked by factors such as CTCF [67], as well as from the borders between A and B compartment regions of chromatin. We next therefore asked whether the loop extrusion process, blocked by different numbers of boundaries, would be expected to generate the patterns of structure variation we observe. We simulated loop extrusion using the chr21:28–30 Mb segment using experimental cohesin binding data from K562 and IMR90 (see Methods). This region has fewer boundaries in K562 (likely due to the portion of the region that is heterochromatic in this cell type). The structure landscapes resulting from a single simulation run show that the cell type with more loop extrusion boundaries (IMR90) would be predicted to have a lower ensemble SSI than the cell type with fewer boundaries if only loop extrusion were at work (S10A and S10B Fig). Once we start the simulation from several different initial conditions, this difference is diminished (S10B Fig) due to variation in structure between different runs. While this result matches our expectations that more boundaries associate with lower ensemble SSI, it does not agree with the experimental observations from these cell types, in which IMR90 actually has the higher ensemble SSI. This suggests that epigenetic state-specific interactions must be considered along with loop extrusion to explain the chromosome variability observed.

The effect of cohesin depletion on structural heterogeneity depends on the underlying epigenetic state of the chromosome region

To further examine the effect of cohesin on the chromosome structural landscape, we analyzed chr21 single-cell structural data from HCT116 cells in wildtype (WT) and cohesin depleted (+Auxin) conditions [37]. This analysis included two different genomic regions (chr21:28–30 Mb and chr21:34–37 Mb) with different underlying structural properties. The jointly constructed chromosome structural landscape for both genomic regions shows that along PC1 the structures are again ordered based on their R_g (S10C and S10D Fig). Thus, the chromosome structural landscape approach orders single-cell structures based on their R_g irrespective of cell-type, genomic regions, and perturbation condition. The WT and cohesin-depleted structural landscapes from the first genomic region mostly overlap with a slight bias for the +Auxin structures in the higher R_g regime (Fig 6C). This suggests that removal of cohesin makes the structures less compact to some extent, as previously observed [68].

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Fig 6. Cohesin depletion alters chromosome structural heterogeneity depending on underlying epigenetic state constraints.

a,b) Population-averaged distance maps of HCT116 chr21:28–30 Mb (a) or chr21:34–37 Mb (b) at 30 kb resolution for wild-type (WT) and cohesin-depleted (+Auxin) conditions based on data obtained from Bintu et al. (37]. Compartment status from Hi-C data at these locations shown above each matrix. c,d) Jointly constructed chromosome structural landscapes of chr21:28–30 Mb (c) or chr21:34–37 Mb (d) of HCT116 for WT and +Auxin conditions. e) Chromosome structural ensemble SSI distributions for different local chromosome regions of HCT116 for WT and +Auxin conditions. 200 single-cell structures are selected randomly each time for resampling.

https://doi.org/10.1371/journal.pcbi.1010392.g006

On the other hand, for the second region, the jointly constructed landscape shows a shift upwards along PC2 along with the shift towards the large R_g regime for cohesin-depleted structures (Fig 6D). In addition, we found that after cohesin removal, both genomic regions exhibit more structural variation at the genomic bin level according to BSI standard deviation (S10E and S10F Fig). When we examine the structural ensemble SSI of these two regions, we see apparently conflicting effects of cohesin removal: the ensemble SSI is decreased in the 28–30 Mb region but mostly unchanged in the 34–37 Mb region (Fig 6E). Note that the initial ensemble SSI of these regions in the WT follows the expected pattern for regions of different lengths. We observed that this divergent ensemble SSI effect corresponds to a difference in the underlying compartment state of the regions. The 28–30 Mb region is divided into A and B compartment segments (which in earlier examples tends to correspond to lower SSI) (Fig 6A) while the 34–37 Mb region is all in the A compartment with no compartment boundaries (which in earlier examples tends to correspond to higher SSI) (Fig 6B).